Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userGeodesic sprays and frozen metrics in rheonomic Lagrange manifolds
59
0
0.0
(
0
)
Authors
Steen Markvorsen
Ask ChatGPT about the research
No Arabic abstract
We define systems of pre-extremals for the energy functional of regular rheonomic Lagrange manifolds and show how they induce well-defined Hamilton orthogonal nets. Such nets have applications in the modelling of e.g. wildfire spread under time- and space-dependent conditions. The time function inherited from such a Hamilton net induces in turn a time-independent Finsler metric - we call it the associated frozen metric. It is simply obtained by inserting the time function from the net into the given Lagrangean. The energy pre-extremals then become ordinary Finsler geodesics of the frozen metric and the Hamilton orthogonality property is preserved during the freeze. We compare our results with previous findings of G. W. Richards concerning his application of Huyghens principle to establish the PDE system for Hamilton orthogonal nets in 2D Randers spaces and also concerning his explicit spray solutions for time-only dependent Randers spaces. We analyze examples of time-dependent 2D Randers spaces with simple, yet non-trivial, Zermelo data; we obtain analytic and numerical solutions to their respective energy pre-extremal equations; and we display details of the resulting (frozen) Hamilton orthogonal nets.
rate research
Read More
Ambrose, Palais and Singer cite{Ambrose} introduced the concept of second order structures on finite dimensional manifolds. Kumar and Viswanath cite{Kumar} extended these results to the category of Banach manifolds. In the present paper all of these results are generalized to a large class of Frechet manifolds. It is proved that the existence of Christoffel and Hessian structures, connections, sprays and dissections are equivalent on those Frechet manifolds which can be considered as projective limits of Banach manifolds. These concepts provide also an alternative way for the study of ordinary differential equations on non-Banach infinite dimensional manifolds. Concrete examples of the structures are provided using direct and flat connections.
We discuss new sufficient conditions under which an affine manifold $(M, abla)$ is geodesically connected. These conditions are shown to be essentially weaker than those discussed in groundbreaking work by Beem and Parker and in recent work by Alexander and Karr, with the added advantage that they yield an elementary proof of the main result.
For compact manifolds with infinite fundamental group we present sufficient topological or metric conditions ensuring the existence of two geometrically distinct closed geodesics. We also show how results about generic Riemannian metrics can be carried over to Finsler metrics.
Let $M$ be a differentiable manifold, $T_xM$ be its tangent space at $xin M$ and $TM={(x,y);xin M;y in T_xM}$ be its tangent bundle. A $C^0$-Finsler structure is a continuous function $F:TM rightarrow mathbb [0,infty)$ such that $F(x,cdot): T_xM rightarrow [0,infty)$ is an asymmetric norm. In this work we introduce the Pontryagin type $C^0$-Finsler structures, which are structures that satisfy the minimum requirements of Pontryagins maximum principle for the problem of minimizing paths. We define the extended geodesic field $mathcal E$ on the slit cotangent bundle $T^ast Mbackslash 0$ of $(M,F)$, which is a generalization of the geodesic spray of Finsler geometry. We study the case where $mathcal E$ is a locally Lipschitz vector field. We show some examples where the geodesics are more naturally represented by $mathcal E$ than by a similar structure on $TM$. Finally we show that the maximum of independent Finsler structures is a Pontryagin type $C^0$-Finsler structure where $mathcal E$ is a locally Lipschitz vector field.
In this paper, we use a Killing form on a Riemannian manifold to construct a class of Finsler metrics. We find equations that characterize Einstein metrics among this class. In particular, we construct a family of Einstein metrics on $S^3$ with ${rm Ric} = 2 F^2$, ${rm Ric}=0$ and ${rm Ric}=- 2 F^2$, respectively. This family of metrics provide an important class of Finsler metrics in dimension three, whose Ricci curvature is a constant, but the flag curvature is not.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
